We follow the method in Ref. [1]} to calculate the GW emission luminosity based on the Newman-Penrose (NP) formalism in Kerr spacetime [2]}. The complex null tetrads \(\lbrace l^\mu , n^\mu , m^\mu , m^{*\mu }\rbrace \)  are defined as,
\(l^{\mu } &=\left[\left(r^{2}+a^{2}\right) / \Delta , 1,0, a / \Delta \right],\\n^{\mu } &=\left[r^{2}+a^{2},-\Delta , 0, a\right] /(2 \Sigma ), \\m^{\mu } &=[i a \sin \theta , 0,1, i / \sin \theta ] \left\bad.\left[\sqrt{2}(r+i a \cos \theta )\right]\right.,\) 
